On Quantum De Rham Cohomology Theory

We define the quantum exterior product ∧h and quantum exterior differential dh on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of dh. We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d by dh and ∧ by ∧h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion. In this note we announce a construction of a deformation of the de Rham complex for any Poisson manifold. In the case of a closed symplectic manifold, its cohomology provides a deformation of the ring structure on the de Rham cohomology. More precisely, on any Poisson manifold, we define a quantum exterior product ∧h of exterior forms, and quantum exterior differential dh, such that dh = 0, and dh is a derivation for ∧h. Here h is an indeterminate. We define the quantum de Rham cohomology as the cohomology of dh. Since dh is a derivation with respect to ∧h, there is an induced quantum multiplication on the quantum de Rham cohomology. This work grew out of our attempt to find a new way to define quantum cohomology, which has recently attracted much attention (see Tian [13] for a survey on this topic, and the introduction of Li-Tian [11] for more recent development). Intuitively, quantum cohomology provides a deformation of the ring structure on the vector space which underlies the de Rham cohomology by counting pseudoholomorphic curves in symplectic manifolds or stable curves in algebraic manifolds. Unlike many cohomology theories in algebraic topology, it is not defined as the cohomology of a graded differential algebra. For the sake of being consistent with other cohomology theories, it would be desirable to be able to do so, even though the applications of quantum cohomology do not really require this property. This is the first motivation for our construction of quantum de Rham cohomology. On the other hand, from the point of view of finding deformations of the ring structure on the de Rham cohomology per se, it is also interesting to see whether deformations of the de Rham complex can provide new deformations of the de Rham cohomology. This is the second motivation to our construction. Received by the editors May 07, 1998. 1991 Mathematics Subject Classification. Primary 53C15, 58A12, 81R05. Authors’ research was supported in part by NSF grants DMS-96-32028 and DMS-95-04925. c ©1999 American Mathematical Society